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Title: Tykhonov well-posedness of a heat transfer problem with unilateral constraints (English)
Author: Sofonea, Mircea
Author: Tarzia, Domingo A.
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940 (print)
ISSN: 1572-9109 (online)
Volume: 67
Issue: 2
Year: 2022
Pages: 167-197
Summary lang: English
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Category: math
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Summary: We consider an elliptic boundary value problem with unilateral constraints and subdifferential boundary conditions. The problem describes the heat transfer in a domain $D\subset \mathbb R^d$ and its weak formulation is in the form of a hemivariational inequality for the temperature field, denoted by $\mathcal P$. We associate to Problem $\mathcal P$ an optimal control problem, denoted by $\mathcal Q$. Then, using appropriate Tykhonov triples, governed by a nonlinear operator $G$ and a convex $\widetilde {K}$, we provide results concerning the well-posedness of problems $\mathcal P$ and $\mathcal Q$. Our main results are Theorems 4.2 and 5.2, together with their corollaries. Their proofs are based on arguments of compactness, lower semicontinuity and pseudomonotonicity. Moreover, we consider three relevant perturbations of the heat transfer boundary valued problem which lead to penalty versions of Problem $\mathcal P$, constructed with particular choices of $G$ and $\widetilde {K}$. We prove that Theorems 4.2 and 5.2 as well as their corollaries can be applied in the study of these problems, in order to obtain various convergence results. (English)
Keyword: heat transfer problem
Keyword: unilateral constraint
Keyword: subdifferential boundary condition
Keyword: hemivariational inequality
Keyword: optimal control
Keyword: Tykhonov well-posedness
Keyword: approximating sequence
Keyword: convergence results
MSC: 35A16
MSC: 35M86
MSC: 49J20
MSC: 49J40
MSC: 49J45
MSC: 49J52
MSC: 80A19
idZBL: Zbl 07511500
idMR: MR4396683
DOI: 10.21136/AM.2021.0172-20
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Date available: 2022-03-25T08:22:13Z
Last updated: 2024-05-06
Stable URL: http://hdl.handle.net/10338.dmlcz/149565
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